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Incompressible limit of a fluid-particle interaction model. (English) Zbl 07332690
Summary: The incompressible limit of the weak solutions to a fluid-particle interaction model is studied in this paper. By using the relative entropy method and refined energy analysis, we show that, for well-prepared initial data, the weak solutions of the compressible fluid-particle interaction model converge to the strong solution of the incompressible Navier-Stokes equations as long as the Mach number goes to zero. Furthermore, the desired convergence rates are also obtained.
35B25 Singular perturbations in context of PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI
[1] Ballew, J.; Trivisa, K., Suitable weak solutions and low stratification singular limit for a fluid particle interaction model, Q. Appl. Math. 70 (2012), 469-494 · Zbl 1418.76045
[2] Ballew, J.; Trivisa, K., Weakly dissipative solutions and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system, Nonlinear Anal., Theory Methods Appl., Ser. A 91 (2013), 1-19 · Zbl 1284.35303
[3] Baranger, C.; Boudin, L.; Jabin, P.-E.; Mancini, S., A modeling of biospray for the upper airways, ESAIM, Proc. 14 (2005), 41-47 · Zbl 1075.92031
[4] Veiga, H. Beirão da, Singular limits in compressible fluid dynamics, Arch. Ration. Mech. Anal. 128 (1994), 313-327 · Zbl 0829.76073
[5] Berres, S.; Bürger, R.; Karlsen, K. H.; Tory, E. M., Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math. 64 (2003), 41-80 · Zbl 1047.35071
[6] Carrillo, J. A.; Goudon, T., Stability and asymptotic analysis of a fluid-particle interaction model, Commun. Partial Differ. Equations 31 (2006), 1349-1379 · Zbl 1105.35088
[7] Carrillo, J. A.; Karper, T.; Trivisa, K., On the dynamics of a fluid-particle interaction model: The bubbling regime, Nonlinear Anal., Theory Methods Appl., Ser. A 74 (2011), 2778-2801 · Zbl 1214.35068
[8] Chemin, J.-Y.; Masmoudi, N., About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal. 33 (2001), 84-112 · Zbl 1007.76003
[9] Chen, Y.; Ding, S.; Wang, W., Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), 5287-5307 · Zbl 1353.35222
[10] Chen, Z.-M.; Zhai, X., Global large solutions and incompressible limit for the compressible Navier-Stokes equations, J. Math. Fluid Mech. 21 (2019), Article ID 26, 23 pages · Zbl 1416.35181
[11] Constantin, P.; Masmoudi, N., Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D, Commun. Math. Phys. 278 (2008), 179-191 · Zbl 1147.35069
[12] Danchin, R.; Mucha, P. B., Compressible Navier-Stokes system: Large solutions and incompressible limit, Adv. Math. 320 (2017), 904-925 · Zbl 1384.35058
[13] Donatelli, D.; Feireisl, E.; Novotný, A., On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions, Discrete Contin. Dyn. Syst., Ser. B 13 (2010), 783-798 · Zbl 1194.35304
[14] Evje, S.; Wen, H.; Zhu, C., On global solutions to the viscous liquid-gas model with unconstrained transition to single-phase flow, Math. Models Methods Appl. Sci. 27 (2017), 323-346 · Zbl 1359.76291
[15] Feireisl, E.; Petcu, M., Stability of strong solutions for a model of incompressible two-phase flow under thermal fluctuations, J. Differ. Equation 267 (2019), 1836-1858 · Zbl 1416.35204
[16] Hsiao, L.; Ju, Q.; Li, F., The incompressible limits of compressible Navier-Stokes equations in the whole space with general initial data, Chin. Ann. Math., Ser. B 30 (2009), 17-26 · Zbl 1181.35171
[17] Huang, B.; Huang, J.; Wen, H., Low Mach number limit of the compressible Navier-StokesSmoluchowski equations in multi-dimensions, J. Math. Phys. 60 (2019), Article ID 061501, 20 pages · Zbl 1444.76086
[18] Huang, F.; Wang, D.; Yuan, D., Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow, Discrete Contin. Dyn. Syst. 39 (2019), 3535-3575 · Zbl 1415.76638
[19] Klainerman, S.; Majda, A., Compressible and incompressible fluids, Commun. Pure Appl. Math. 35 (1982), 629-651 · Zbl 0478.76091
[20] Lin, C.-K., On the incompressible limit of the compressible Navier-Stokes equations, Commun. Partial Differ. Equations 20 (1995), 677-707 · Zbl 0816.35105
[21] Lions, P.-L., Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models, Oxford Lecture Series in Mathematics and Its Applications 10. Clarendon Press, Oxford (1998) · Zbl 0908.76004
[22] Lions, P.-L.; Masmoudi, N., Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., IX. Sér 77 (1998), 585-627 · Zbl 0909.35101
[23] Masmoudi, N., Incompressible, inviscid limit of the compressible Navier-Stokes system, Ann. Inst. Henri Poincaré, Anal. Non linéaire 18 (2001), 199-224 · Zbl 0991.35058
[24] Ou, Y., Incompressible limits of the Navier-Stokes equations for all time, J. Differ. Equations 247 (2009), 3295-3314 · Zbl 1181.35177
[25] Vauchelet, N.; Zatorska, E., Incompressible limit of the Navier-Stokes model with a growth term, Nonlinear Anal., Theory Methods Appl., Ser. A 163 (2017), 34-59 · Zbl 1370.35234
[26] Vinkovic, I.; Aguirre, C.; Simoëns, S.; Gorokhovski, M., Large eddy simulation of droplet dispersion for inhomogeneous turbulent wall flow, Int. J. Multiphase Flow 32 (2006), 344-364 · Zbl 1135.76570
[27] Williams, F. A., Spray combustion and atomization, Phys. Fluids 1 (1958), 541-545 · Zbl 0086.41102
[28] Williams, F. A., Combustion Theory, CRC Press, Boca Raton (1985)
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